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Segregation kinetics and surface phase transitions
342
Surface Science 200 (1988) 342-353 North-Holland, Amsterdam
SEGREGATION KINETICS AND SURFACE PHASE TRANSITIONS M. MILITZER and J. WIETING Akademie der Wissenschaflen der DDR, Zentralinstitut ]'fir Festk~r?erphysik und Werkstofforschung, DDR-8027 Dresden, German Dem. Rep. Received 16 May 1987; accepted for publication 29 November 1987
A phenomenologicai approach of segregation kinetics in binary systems including structural surface phase transitions with a rather long nucleation time is extended to ternary systems introducing such a structural transition into the ternary regular solution model. Both structures are characterized by different segregation energies and saturation values. A comparison with surface segregation measurements by AES of Pb in Ag and Sn in Fe as well as in Fe-6at%Si shows good agreement. In the latter case a proper description of the segregation competitian between Si and Sn is achieved.
1. Introduction
The segregation to surfaces as well as grain boundaries in binary and ternary alloys has received widespread attention in the last decade [1-4]. Measurements of segregation kinetics showing a t 1/2 law confirm an enrichment controlled by bulk diffusion [1,5-8]. Recently, in some cases of segregation to low index crystallographic planes deviations from this behaviour have been observed, since a two-dimensional (2D) structural phase transition takes place in the surface layer [7-11]. Because of a rather long nucleation time this interracial phase transition may become a rate determining step of the whole segregation process. In a recent paper [12] we proposed a phenomenological approach of segregation in binary systems including structural transitions due to the segregating atoms themselves. In the present contribution this approach is extended to ternary systems, where 2D structural transitions may occur due to different structures formed by competitive segregants. After introducing the corresponding model it is applied to experimental results by Zhou et al. [13] of tin segregation in an iron-silicon alloy. For comparison a short discussion of segregation in binary iron and silver alloys is given, too. 2. Model The basic equations describing the kinetics of segregation were derived in a previous paper [14]. In the case of constant temperature T and homogeneous 0030-6028/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
M. Militzer, J. Wieting / Segregation kinetics and surface phase transitions
343
initial distribution c o of segregant i one obtains
ci(t)=c(O)+2c°(Dit/~$2cr)l/2-(Di/~2cr)
1/2
/'t
,
,
joqi(t )(t-t )
- 1/2
dt',
(1)
where c~(t) denotes the interfacial coverage at time t, ~ the layer thickness, D, the respective bulk diffusion coefficient and qi(t) the concentration in the matrix layer adjacent to the interface. Concentration as well as coverage are normalized to the number of available sites in the considered region. Local equilibrium is assumed between c~ and q~ in a regular solution model as proposed by Guttmann [15]. Including site competition the following equation holds: c, e x p ( - G~X)/kT), (2) q i = x x - ~'-'.cj J where x x is the saturation coverage of surface structure h and G~x) the respective free energy of segregation, G~x'
=
E/(x'+ E
J
(x),. Olij ~'j,
(3)
where the parameters utij ~(x) represent nearest neighbour interaction of segregating species in the boundary layer. Although eq. (1) suggests a bulk-diffusion-controlled enrichment rate, effects of interfacial r:actions are considered within local equilibrium conditions qi(t). To account for a considerable nucleation time as observed for 2D structural transitions connected with the existence of metastable surface states the pure Guttmann relation has to be modified, i.e. a proper estimation of surface structure X which governs q,(t) is required. In order to solve this problem a more detailed consideration of segregation to well defined surfaces with respect to their energy levels is necessary. First, we shortly remember the situation hi binary systems with one segregant only. At the beginning foreign atoms segregate or adsorb at energetically favoured surface sites ( E ° ) ) . Consequently, an ordered arrangement being commensurate to the surface structure is built up. Moreover, segregating atoms may form a second, frequently incommensurate structure involving more sites than the first one. Although the binding energy per atom must be somewhat smaller in the second structure (E(2)), it becomes favourable at l~gher enrichment states. But the formation of this new phase requires at least locally a strong oversaturation of the first ordered structure, which may be realized by . . . . . . . ,:~.....t .~aa;,;,~,,~i ;nt,~r~titial-!ike sites in the commensurate surface structure with significantly diminished free energy of segregation E (°~. Assuming two foreign atoms are to occupy one site of the well ordered structure an average segregation free energy of E* = ½( E " ' + E
(4)
344
M. Militzer, J. Wieting / Segregation kinetics and surface phase transitions
CT Xo I
C2
!
Ccrif Fig. 1. Surface free energies of two structures with different satt ration values (x 0, 1) and E (2) < E ° ) in an effective ideal solution model versus coverage c at T = constant in a binary system (schematically).
is evaluated, if both sites are occupied. In this way the coverage within structure 1 may exceed the saturation value x 0 (x 0 < 1) with respect to the energetically favoured sites. Note that the saturation value of the second structure is normalized to 1. The concentration c* of foreign atoms in the segregation state E * is determined from the minimum of surface free energy f~ of structure 1 with respect to c* [12] Oc* - o,
> o.
(5)
Fig. 1 shows schematically the surface free energy versus coverage c of segregating atoms in an effective ideal solution model with E tl) > E t2). The reversed case (E (2~ > E (1~) is not of interest, since structure 2 would dominate
M. Mifitzer,J. Wieting/ Segregationkineticsand surfacephasetransitions
345
in the whole concentration range. The interracial equilibrium of both structures is determined by the equality of the chemical potentials which is equivalent to the well known tangential construction. The corresponding coverages are c~ and c 2, respectively. But reaching c~ is not sufficient to nucleate structure 2, which indeed requires a strong oversaturation of phase 1. Consequently, the interfacial coverage has to accomplish at least locally a higher value cerit, which is determined by the equality of surface free energies of both structures,
=A(Ccn,).
(6)
Therefore, q(t) is given by structure 1 as long as c(t)< cerit holds. The structural transition takes place in the range above cer~t and below c 2, where structure 1 (c~) is in equilibrium with structure 2 (c 2). If c(t) exceeds c 2, only structure 2 governs q(t). Thus, the local equilibrium conditions are written
¢(t) - c * ( t ) Xo-C(t)+c*(t)/2 q(t) =
exp(_Et,,/kT )
¢,-c~ exp(_Et,)/kT ) X o - c~ + c~/2 _
c--L-E e x p ( - E t E ) / k T ) 1 -c
(c(t)
< c,:rit),
(c¢,.
(7)
2
c( t ) exp(- E 2'/kT ) 1 -c(t)
(¢(I)>¢2),
as shown in ref. [12]. Conditions (7) are used to evaluate the segregation kinetics in respective binary systems. It directly comes from (7), that such a structural transition may only occur, if c o is sufficiently high, i.e. ¢crit -- ¢crit
exp(-E"'/kT)
(8)
cO > qcrit= Xo _ Ccrit + Cc.rit/2 is valid. The description of the analogous case in ternary systems is rather similar. In contrast to binary systems there is a larger probability for commensurate-commensurate phase transitions. Let A be the element with the smaller interfacial activity and the higher mobility L i = c°D)/2 compared ...;,~. D t r ~. r ~ ~~,,A ~nh, ,hit ,-~S,~ it. nf interest here~ since otherwise no wtui Ju, I L , A . ~ ~ t - , B ] , --,~ v~,--7 . . . . . . . . . . . . . . competition effect is observed [14]. Then at first the segregation structure of A is formed. Later on A is replaced by B. The surface structure transforms, if the B coverage has reached a certain level. The corresponding structure leads to an energetically favoured interface state being not only caused by a sufficiently high segregation free energy of B, but even more by a higher enrichment ratio because of a larger saturation value x 2 > x~.
M. Militzer, J. Wieting / Segregation kinetics and surface phase transitions
346
A proper description of this transition requires an exact knowledge of the respective structures and cannot be given in a general form. In order to show some principal features we want to discuss a transition where the first arrangement already represents a substructure of the new phase. Thus, occupation of additional sites in the first structure and forming the second phase are equivalent and require an enrichment of B in the first phase exceeding the value which is thermodynamically expected from the equality of the respective chemical potentials. Otherwise the number of B atoms in the surface region would be too small for the nucleation of the second structure. According to binary systems the transition will occur, if both surface free energies are equal. Assuming x2 = 1 and a regular solution model, the respective surface free energy of structure Jk (h = 1, 2) is given by i
+kT[~ic i.
i,j
In c i + ( x x -
~ci)i
ln(xx - )-'c~, ) i - x x In x x ,
(9,
where terms contributing to both energies are omitted. Provided the interactions a ~ ) are sufficiently strong, miscibility gaps may occur within structure ~,. To evaluate the value of fx for this case the corresponding demixing relations have to be taken into account [16]. Up to the transition at t = t. connected with a distinct segregation ratio cB/cA local equilibrium (2) is determined by structure 1 whereas for t > t. structure 2 governs (2).
3. Results and discussion
3.1. Lead segregationin silver Rolland and Bernardini [7] investigated the segregation kinetics of lead to the (111) surface of a Ag-0.5at%Pb single crystal at temperatures of 533-587 K by means of Auger electron spectroscopic (AES) measurements. Fig. 2a shows the experimental res:dts. An order-order transition of a commensurate-incommensurate type is clearly observed at 575 as well as 564 K. An apparent saturation plateau is formed at about c = 0 . 5 with a p(V~× V/~)R30 o structure [9]. In contrast the final saturation (c = 1) corresponds to 9.42 × 1014 lead atoms per cm 2 arranged in a close packed layer. Thus x 0 = 0.5 holds and 8 = 1.6 A is assumed. The bulk diffusion coefficient of Pb in As, D~b= 1.5 e x p ( - 1 . 6 7
eV/kT)
cm 2 s - l ,
was determined in ref. [7]. in order to enable a more quantitative comparison
M. Mifitzer, J. Wieting / Segregation kinetics and surfacephase transitions cPbI
347
AES measurement
o.a o. o.t o.
a
I
theory
;17fJ '" io
is tlks
Fig. 2. Segregation kinetics of Pb to the (111) surface in A g - 0 . 5 a t % P b ; (a) results of A E S m e a s u r e m e n t s a f t e r RoUand a n d B e r n a r d i n i [7]; (b) present calculations with x 0 = 0.5, E0tl) = 0.52 eV, A E = 0.13 eV, A E , = 0.30 eV, S = 5 × 10 - 4 e V / K , Dvb = 1.5 exp( -- 1.67 eV/kT) cm 2 s - !.
of model calculations with AES results in the whole temperature range under study, the actual temperature dependence of the segregation free energy, E (x) = Eto x) + Ts(X),
(10)
has to be taken into account, i.e. the excess entropy of segregation S (x) describing the differences of vibrational entropies of the segregant in the bulk and the surface is treated explicitly. Commonly only an average contribution of TS (x) is considered because of the smallness of S (x). Its value is expected somewhat above 10 -4 e V / K [1,2]. In the given example the excess entropy of segregation is approximated by S (1) -- 8 (2) --" S *
= S = 5 x 10 -4
eV/K.
Assuming a segregation free energy of about 0.8 eV, E0~1) is estimated to be 0.52 eV. The results of model calculations with AE = E (1) - E (2) =0.13 eV and A E . = E (l) - E* = 0.30 eV are presented in fig. 2b. It shows rather good agreement with the AES measurements. At higher temperatures (587 K) the cqfichment is more or less undisturbed and follows a parabolic law up to final
348
M. Militzer, J. Wieting / Segregation kinetics and surface phase transitions
|
tO" Cpb
/~ E=OlleV
E=O.13eV
E=O.135eV
0.6 0.~ 0.2-
~i~
15
2'0 t/ks
Fig. 3. Influence of slightly altered values of A E on the segregation kinetics with a 2D structural transition shown for Pb in Ag-0.5at%Pb at 564 K.
saturation. Because of the large mobility of segregating species in the bulk, the plateau indicating the saturation of structure 1 is very short and hardly observable in experiments. With decreasing temperature diffusion is slowed down. Therefore, reaching Ccnt is more and more delayed. Significant plateaus at 575 as well as at 564 K indicate the saturation of the first structure accompanied by a rather high value of q(t) tending to c °. Reaching Ccrit the new structure is formed and q(t) quickly drops down leading to a significant increase of enrichment rate. At 533 K the annealing time ( - 300 min) is not sufficient to achieve a structural transition. Furthermore, the experiment suggests an apparent saturation at only 1/3 which could be a result of lead precipitation in the interior lowering the effective c o with annealing time, since the solubility limit of Pb in Ag is below 0.5 at% at 533 K [17]. Consequently, nucleation of the second structure may be generally suppressed at lower temperatures. Moreover, the plateau lengths are rather sensitive to changes in A E and A E., respectively. This is demonstrated in fig. 3, where the kinetics at 564 K is shown for three slightly altered values of A E: Since ('crit . i n c r e ~ w i t h A P a n d qcrit tends closer to co a significantly enlarged delay in formation of the new phase results, although the variation of AE is only in the range of 0.01 eV < k T. Therefore, a respective transition width is expected leading to a smooth increase of further enrichment as observed in AES measurements in contrast to the evaluations with fixed AE. Consequently, the second enrichment rate cannot be used in order to estimate diffusion coefficient and bulk concentration, respectively, as widely done for an undisturbed segregation, e.g. the first enrichment period in the given case [7].
M. MiMzer, J. Wieting / Segregation kSnetics and surface phase transitions
349
3.2. Tin segregation in iron alloys
Similar results as observed for Pb in Ag are measured in Fe-l.9atToSn [8,10]. A n order-disorder transition of segregated tin atoms was found at the (100) surface. A calibration yields surface coverages of 6.1 x 1014 atoms per cm 2 for the ordered c(2 x 2) structure as well as 1.4 x 10 is atoms per cm 2 for final saturation. Therefore, x 0 = 0.45 is evaluated. The. experimental result at T = 858 K indicates an enrichment time of the first structure comparable to the period of the plateau at the apparent saturation level. Fig. 4 shows the result of the given approach with E °> = 1.0 eV, AE = 0.16 eV, A E , = 0.40 eV, where the value of E tl~ is chosen according to suggestions from experiment [18]. The model calculation with these parameters is in agreement with the experimental observation. For a more quantitative comparison we refer to the tin segregation in Fe-6.2at%Si-0.03at%Sn, wbSch is a widely used soft-magnetic material. Zhou et al. [13] investigated the segregation competition of Sn and Si on the (100) surface at 873 K by AES and LEED. Because of Cs°i>> cOn at first silicon segregates and reaches saturation corresponding to the equilibrium in a binary F e - S i alloy at half a monolayer, where a c(2 × 2) structure is formed. Later on tin displaces silicon from the surface, since the interracial activity of tin is considerably higher. The segregation free energy of silicon only amounts to 0.64 eV at the given temperature [2]. The final saturation of tin segregation corresponds to a p(1 × 1) structure being equivalent to a tin surface coverage of one monolayer. Because of different saturation values the competition between both segregants cannot be described in a simple regular solution
1.o CSn
0.8 0.6.
n/.
5'oo
1o'oo
15oo
2ooo os, t 6~
Fig. 4. Calculated surface segregation kinetics of Sn in Fe including a structural transition (x o = 0.45, E t~ =1.0 eV, AE = 0.16 eV, AE. = 0.40 eV, c° = 0.019, T= 858 K).
M. Militzer, J. Wieting / Segregation kinetics and surface phase transitions
350
model. Indeed the latter has to be connected with a structural transition in the surface layer as already proposed. For convenience furthar approximations are possible. Because of its high mobility silicon is in a current equilibrium (qsi = Cs°i) during the replacement. Then tin diffusion determines the competition kinetics. In order to estimate proper local equilibrium conditions for tin segregation a strong repulsive interaction ,,o) ~ ' S i S n between segregated Si and Sn has to be taken into account in the silicon-rich structure 1, whereas self-interactions a~f ) are neglected. The silicon-tin interaction in iron, =
- (
-- , o
-- ,o
os. ) ,
(11)
is expected to be strongly negative from phase diagrams of the respective binary systems. The regular miscibility parameters are given by
toij = Z [ ~.ij - I ( ~.ii -~ ¢£jj)] ,
(12)
where Z denotes the coordination number and % < 0 the pair interactions. Because of the lack of any miscibility between silicon and tin t%iSn > 0 holds, whereas the formation of intermetallic compounds yields evesi ~<0 as well as ~0veS. < 0. Thus a strong repulsive interaction, e.g. ,~(~) "~SiSn --" - 0 . 5 eV, is assumed leading to an extended miscibility gap in the first surface structure at the given temperature. Demixing already occurs for Cs, < 0.01 provided, Csi is sufficiently high. Therefore, the local equilibrium condition of Sn is given up to the structural transition by qSn(t)
=
exp
kT
~'Sn
=q0,
(13)
as shown in ref. [12]. A sufficient tin enrichment enables the transition c(2 × 2) --* p(1 x 1). For structure 2, p(1 x 1), an effective ideal solution model is used, where the strong Si-Sn repulsion is taken into account by an extremely small segregation energy of silicon Es(2) << E Si(1)' whereas ~-t2)> r.tl) "Sn J-'Sn is expected. Thus tin segregation is undisturbed by silicon in the second phase nucleating at t, and qs,(t) is then given by qsn(t) = q ( t ) =
Cs"(t)
1 - Y'.cj(t)
e x p ( E(s2'/kT)
(14)
J
with Csi (t) = [ 1 - Cs. (t)] c°i exp( E(s2'/k T) 1 + c°i exp( E(s2'/k T ) '
(I 5)
characterizing the current equilibrium of silicon in the new structure. Because of a high enrichment ratio of tin in this second phase q ( t ) << cO,
M. Militzer, J. Wieting / Segregation kinetics and surface phase transitions
351
X h.-~',,,,~',='
Fe - 6.2%Si - 0,03%Sn 873K
c,° 1
0#.
~ Si
i,
I ~t/
Sn~~x
zx z~
o.~-
~
x
1o
~ - r -Io~ " t /rnin
Fig. 5. C o m p a r i s o n of the calculated segregation kinetics to the (100) surface in an F e - 6 . 2 a t % S i - 0 . 0 3 a t % S n single crystal at 873 K with A E S measurements (Si: (zx), Sn: ( x ) ) of Z hou et al. [13]; ~t~°),.si= 0.64 eV, E 12)si= 0.1 eV, ---s.~'°)= 1.07 eV, E t2}sn= 1.5 eV, D~.8 -2. = 66).
holds up to final saturation. Therefore, segregation equation (1) is approximated by 2(C~n--q°)(Dsnt//~
Csn(t ) =
2
~r)
!/2
,
t
2(Dsn//82~r),/2(cOnt,/2_ q°t~/2),
(16)
t > t,,
where according to the experiment Csn(0) = 0 is introduced. A fit with the experimental result of Zhou et al. yields Dsn/82= 66 which agrees well with expectations from other experimental results on tin diffusion in ferromagnetic a-Fe [8,19]. It was already mentioned that the segregation free energy of tin in pure iron amounts to about 1.0 eV, in iron-silicon alloys it is expected to be even higher. Using ~'t~) 1.07 eV, L'Sn ~,t2) 1.5 eV as well as Es~i2~= 0.1 eV good agreement with the AES measurement by Zhou et ai. is achieved as shown in fig. 5. For comparison the AES results given in relative peak height ratios (PHR~) with respect to the 703 eV iron signal are converted into coverages by the linear relation ~Sn
"-"
- - "
c, = h i PHIL,
(17)
where the calibration constants h i are deduced from LEED measurements of the corresponding structures
PHRsi.max= A
PHRsn.max
A
¢Si,max "- 0 . 5 , m
= ¢Sn,max --
1
.
For silicon a background signal according to c°i is taken into account.
M. Mifitzer, J. Wieting / Segregation kinetics and surf,zce phase transitions
352
Both experimental results and model calculations show a retarded enrichment rate for tin at the beginning of the segregation process due to the strong repulsion with silicor., which increases significantly during the formation of the second structure. It should be pointed out that the replacement of Si by Sn only occurs, if the tin bulk concentration exceeds q0. Consequently, a proper knowledge of this value is useful in a detailed consideration of the corresponding segregation competition. Moreover, it is worth noting th~,t tin segregation structures in iron alloys completely change with varying bulk composition. Whereas in F e - S n a c(2 x 2) structure and finally a disordered one with a tin coverage of more than one monolayer is found, in Fe-Si tin segregation forms a p(1 x 1) structure at (100) planes.
4. Conduding remarks An improved phenomenological approach including 2D structural transitions explains successfully segregation kinetics in binary as well as ternary alloys. Especially in the latter a more quantitative description of competition effects may be achieved allowing a detailed discussion of conditions being necessary for segregation dominance of distinct species. Moreover, it is possible to deal not only with segregation in metals but also in ceramics as well as semiconductors within the presented theory. Frequently, there is a lack in the knowledge of exact absolute values of segregation parameters (E~tx), aij(x) ). Applying the given approach to respective experimental results it enables a determination of relations between these parameters, e.g. in binary systems the energy differences AE of different structures as well as in ternary systems the relation E~/X)= E~ ~) - E ) x) for competitive segregants. Thus starting from one known value, several segregation parameters may be deduced by this method.
References [11 S. Hofmann, in: Scanning Electron Microscopy/1985, Ed. O. Johari (AMF O'Hare, Chicago, 1985) p. 1071. f'~ll tr..l
l,J,
'm~,,.Jlt~ul~.l.,,
~,JL~.,t,,,I
l%~,,.0..Jr
~.LJt..P,J~
•
l~v.
[31 M.P. Seah, in: Practical Surface Analysis by Auger and X-ray Photoelectron Spectroscopy, Eds. D. Briggs and M.P. Seah (Wiley, New York, 1983) p. 247. 14l C.3. MeMahon, Jr. and L. Marchut, J. Vacuum Sci. Technol. 15 (1978) 450. [51 M.P. Seah and C. Lea, Phil. Mag. 35 (1977) 213. 161 G. Luckman, L.R. Adler and W.R. Graham, Surface Sci. 121 (1982) 61. [71 A. Rolland and J. Bernardini, Scripta Met. 19 (1985) 839. [81 K. Hennesen, H. Keller and H. Viefhaus, Scripta Met. 18 (1984) 1319. [91 A. Rolland, J. Bernardini and M.G. Barthes-Labrousse, Surface Sci. 143 (1984) 579. [101 H. Viefhaus and M. Riisenberg, Surface Sci. 159 (1985) 1.
M. Mifitzer, J. Wieting / Segregation kinetics and surface phase transitions [11] [12] [13] [14] [15] [16] [17]
353
F. Bezuidenhout, J. Du Plessis and P.E. Viljoen, Surface Sci. 171 (1986) 392. M. Militzer and J. Wieting, Acta Met. 35 (1987) 2765. Y.X. Zhou, C.J. McMahon, Jr. and E.W. Plummer, J. Vacuum Sci. Technol. A2 (10~4) 1118. M. Militzer and J. Wieting, Acta Met. 34 (1986) 1229. M. Guttmann, Surface Sci. 53 (1975) 213. P. Haasen, Physikalische Metallkunde (Akademie-Vedag, Berlin, 1985). M. Hansen and K. Anderko, Constitution of BinaD" Alloys (McGraw-Hill, New York, 1958) p. 40. [18] M.P. Seah and C. Lea, Phil. Mag. 31 (1975) 627. [19] W. Arabczyk, M. Militzer, H.J. Miissig and J. Wieting, Scripta Met. 20 (1986) 1549.
Surface Science 200 (1988) 342-353 North-Holland, Amsterdam
SEGREGATION KINETICS AND SURFACE PHASE TRANSITIONS M. MILITZER and J. WIETING Akademie der Wissenschaflen der DDR, Zentralinstitut ]'fir Festk~r?erphysik und Werkstofforschung, DDR-8027 Dresden, German Dem. Rep. Received 16 May 1987; accepted for publication 29 November 1987
A phenomenologicai approach of segregation kinetics in binary systems including structural surface phase transitions with a rather long nucleation time is extended to ternary systems introducing such a structural transition into the ternary regular solution model. Both structures are characterized by different segregation energies and saturation values. A comparison with surface segregation measurements by AES of Pb in Ag and Sn in Fe as well as in Fe-6at%Si shows good agreement. In the latter case a proper description of the segregation competitian between Si and Sn is achieved.
1. Introduction
The segregation to surfaces as well as grain boundaries in binary and ternary alloys has received widespread attention in the last decade [1-4]. Measurements of segregation kinetics showing a t 1/2 law confirm an enrichment controlled by bulk diffusion [1,5-8]. Recently, in some cases of segregation to low index crystallographic planes deviations from this behaviour have been observed, since a two-dimensional (2D) structural phase transition takes place in the surface layer [7-11]. Because of a rather long nucleation time this interracial phase transition may become a rate determining step of the whole segregation process. In a recent paper [12] we proposed a phenomenological approach of segregation in binary systems including structural transitions due to the segregating atoms themselves. In the present contribution this approach is extended to ternary systems, where 2D structural transitions may occur due to different structures formed by competitive segregants. After introducing the corresponding model it is applied to experimental results by Zhou et al. [13] of tin segregation in an iron-silicon alloy. For comparison a short discussion of segregation in binary iron and silver alloys is given, too. 2. Model The basic equations describing the kinetics of segregation were derived in a previous paper [14]. In the case of constant temperature T and homogeneous 0030-6028/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
M. Militzer, J. Wieting / Segregation kinetics and surface phase transitions
343
initial distribution c o of segregant i one obtains
ci(t)=c(O)+2c°(Dit/~$2cr)l/2-(Di/~2cr)
1/2
/'t
,
,
joqi(t )(t-t )
- 1/2
dt',
(1)
where c~(t) denotes the interfacial coverage at time t, ~ the layer thickness, D, the respective bulk diffusion coefficient and qi(t) the concentration in the matrix layer adjacent to the interface. Concentration as well as coverage are normalized to the number of available sites in the considered region. Local equilibrium is assumed between c~ and q~ in a regular solution model as proposed by Guttmann [15]. Including site competition the following equation holds: c, e x p ( - G~X)/kT), (2) q i = x x - ~'-'.cj J where x x is the saturation coverage of surface structure h and G~x) the respective free energy of segregation, G~x'
=
E/(x'+ E
J
(x),. Olij ~'j,
(3)
where the parameters utij ~(x) represent nearest neighbour interaction of segregating species in the boundary layer. Although eq. (1) suggests a bulk-diffusion-controlled enrichment rate, effects of interfacial r:actions are considered within local equilibrium conditions qi(t). To account for a considerable nucleation time as observed for 2D structural transitions connected with the existence of metastable surface states the pure Guttmann relation has to be modified, i.e. a proper estimation of surface structure X which governs q,(t) is required. In order to solve this problem a more detailed consideration of segregation to well defined surfaces with respect to their energy levels is necessary. First, we shortly remember the situation hi binary systems with one segregant only. At the beginning foreign atoms segregate or adsorb at energetically favoured surface sites ( E ° ) ) . Consequently, an ordered arrangement being commensurate to the surface structure is built up. Moreover, segregating atoms may form a second, frequently incommensurate structure involving more sites than the first one. Although the binding energy per atom must be somewhat smaller in the second structure (E(2)), it becomes favourable at l~gher enrichment states. But the formation of this new phase requires at least locally a strong oversaturation of the first ordered structure, which may be realized by . . . . . . . ,:~.....t .~aa;,;,~,,~i ;nt,~r~titial-!ike sites in the commensurate surface structure with significantly diminished free energy of segregation E (°~. Assuming two foreign atoms are to occupy one site of the well ordered structure an average segregation free energy of E* = ½( E " ' + E
(4)
344
M. Militzer, J. Wieting / Segregation kinetics and surface phase transitions
CT Xo I
C2
!
Ccrif Fig. 1. Surface free energies of two structures with different satt ration values (x 0, 1) and E (2) < E ° ) in an effective ideal solution model versus coverage c at T = constant in a binary system (schematically).
is evaluated, if both sites are occupied. In this way the coverage within structure 1 may exceed the saturation value x 0 (x 0 < 1) with respect to the energetically favoured sites. Note that the saturation value of the second structure is normalized to 1. The concentration c* of foreign atoms in the segregation state E * is determined from the minimum of surface free energy f~ of structure 1 with respect to c* [12] Oc* - o,
> o.
(5)
Fig. 1 shows schematically the surface free energy versus coverage c of segregating atoms in an effective ideal solution model with E tl) > E t2). The reversed case (E (2~ > E (1~) is not of interest, since structure 2 would dominate
M. Mifitzer,J. Wieting/ Segregationkineticsand surfacephasetransitions
345
in the whole concentration range. The interracial equilibrium of both structures is determined by the equality of the chemical potentials which is equivalent to the well known tangential construction. The corresponding coverages are c~ and c 2, respectively. But reaching c~ is not sufficient to nucleate structure 2, which indeed requires a strong oversaturation of phase 1. Consequently, the interfacial coverage has to accomplish at least locally a higher value cerit, which is determined by the equality of surface free energies of both structures,
=A(Ccn,).
(6)
Therefore, q(t) is given by structure 1 as long as c(t)< cerit holds. The structural transition takes place in the range above cer~t and below c 2, where structure 1 (c~) is in equilibrium with structure 2 (c 2). If c(t) exceeds c 2, only structure 2 governs q(t). Thus, the local equilibrium conditions are written
¢(t) - c * ( t ) Xo-C(t)+c*(t)/2 q(t) =
exp(_Et,,/kT )
¢,-c~ exp(_Et,)/kT ) X o - c~ + c~/2 _
c--L-E e x p ( - E t E ) / k T ) 1 -c
(c(t)
< c,:rit),
(c¢,.
(7)
2
c( t ) exp(- E 2'/kT ) 1 -c(t)
(¢(I)>¢2),
as shown in ref. [12]. Conditions (7) are used to evaluate the segregation kinetics in respective binary systems. It directly comes from (7), that such a structural transition may only occur, if c o is sufficiently high, i.e. ¢crit -- ¢crit
exp(-E"'/kT)
(8)
cO > qcrit= Xo _ Ccrit + Cc.rit/2 is valid. The description of the analogous case in ternary systems is rather similar. In contrast to binary systems there is a larger probability for commensurate-commensurate phase transitions. Let A be the element with the smaller interfacial activity and the higher mobility L i = c°D)/2 compared ...;,~. D t r ~. r ~ ~~,,A ~nh, ,hit ,-~S,~ it. nf interest here~ since otherwise no wtui Ju, I L , A . ~ ~ t - , B ] , --,~ v~,--7 . . . . . . . . . . . . . . competition effect is observed [14]. Then at first the segregation structure of A is formed. Later on A is replaced by B. The surface structure transforms, if the B coverage has reached a certain level. The corresponding structure leads to an energetically favoured interface state being not only caused by a sufficiently high segregation free energy of B, but even more by a higher enrichment ratio because of a larger saturation value x 2 > x~.
M. Militzer, J. Wieting / Segregation kinetics and surface phase transitions
346
A proper description of this transition requires an exact knowledge of the respective structures and cannot be given in a general form. In order to show some principal features we want to discuss a transition where the first arrangement already represents a substructure of the new phase. Thus, occupation of additional sites in the first structure and forming the second phase are equivalent and require an enrichment of B in the first phase exceeding the value which is thermodynamically expected from the equality of the respective chemical potentials. Otherwise the number of B atoms in the surface region would be too small for the nucleation of the second structure. According to binary systems the transition will occur, if both surface free energies are equal. Assuming x2 = 1 and a regular solution model, the respective surface free energy of structure Jk (h = 1, 2) is given by i
+kT[~ic i.
i,j
In c i + ( x x -
~ci)i
ln(xx - )-'c~, ) i - x x In x x ,
(9,
where terms contributing to both energies are omitted. Provided the interactions a ~ ) are sufficiently strong, miscibility gaps may occur within structure ~,. To evaluate the value of fx for this case the corresponding demixing relations have to be taken into account [16]. Up to the transition at t = t. connected with a distinct segregation ratio cB/cA local equilibrium (2) is determined by structure 1 whereas for t > t. structure 2 governs (2).
3. Results and discussion
3.1. Lead segregationin silver Rolland and Bernardini [7] investigated the segregation kinetics of lead to the (111) surface of a Ag-0.5at%Pb single crystal at temperatures of 533-587 K by means of Auger electron spectroscopic (AES) measurements. Fig. 2a shows the experimental res:dts. An order-order transition of a commensurate-incommensurate type is clearly observed at 575 as well as 564 K. An apparent saturation plateau is formed at about c = 0 . 5 with a p(V~× V/~)R30 o structure [9]. In contrast the final saturation (c = 1) corresponds to 9.42 × 1014 lead atoms per cm 2 arranged in a close packed layer. Thus x 0 = 0.5 holds and 8 = 1.6 A is assumed. The bulk diffusion coefficient of Pb in As, D~b= 1.5 e x p ( - 1 . 6 7
eV/kT)
cm 2 s - l ,
was determined in ref. [7]. in order to enable a more quantitative comparison
M. Mifitzer, J. Wieting / Segregation kinetics and surfacephase transitions cPbI
347
AES measurement
o.a o. o.t o.
a
I
theory
;17fJ '" io
is tlks
Fig. 2. Segregation kinetics of Pb to the (111) surface in A g - 0 . 5 a t % P b ; (a) results of A E S m e a s u r e m e n t s a f t e r RoUand a n d B e r n a r d i n i [7]; (b) present calculations with x 0 = 0.5, E0tl) = 0.52 eV, A E = 0.13 eV, A E , = 0.30 eV, S = 5 × 10 - 4 e V / K , Dvb = 1.5 exp( -- 1.67 eV/kT) cm 2 s - !.
of model calculations with AES results in the whole temperature range under study, the actual temperature dependence of the segregation free energy, E (x) = Eto x) + Ts(X),
(10)
has to be taken into account, i.e. the excess entropy of segregation S (x) describing the differences of vibrational entropies of the segregant in the bulk and the surface is treated explicitly. Commonly only an average contribution of TS (x) is considered because of the smallness of S (x). Its value is expected somewhat above 10 -4 e V / K [1,2]. In the given example the excess entropy of segregation is approximated by S (1) -- 8 (2) --" S *
= S = 5 x 10 -4
eV/K.
Assuming a segregation free energy of about 0.8 eV, E0~1) is estimated to be 0.52 eV. The results of model calculations with AE = E (1) - E (2) =0.13 eV and A E . = E (l) - E* = 0.30 eV are presented in fig. 2b. It shows rather good agreement with the AES measurements. At higher temperatures (587 K) the cqfichment is more or less undisturbed and follows a parabolic law up to final
348
M. Militzer, J. Wieting / Segregation kinetics and surface phase transitions
|
tO" Cpb
/~ E=OlleV
E=O.13eV
E=O.135eV
0.6 0.~ 0.2-
~i~
15
2'0 t/ks
Fig. 3. Influence of slightly altered values of A E on the segregation kinetics with a 2D structural transition shown for Pb in Ag-0.5at%Pb at 564 K.
saturation. Because of the large mobility of segregating species in the bulk, the plateau indicating the saturation of structure 1 is very short and hardly observable in experiments. With decreasing temperature diffusion is slowed down. Therefore, reaching Ccnt is more and more delayed. Significant plateaus at 575 as well as at 564 K indicate the saturation of the first structure accompanied by a rather high value of q(t) tending to c °. Reaching Ccrit the new structure is formed and q(t) quickly drops down leading to a significant increase of enrichment rate. At 533 K the annealing time ( - 300 min) is not sufficient to achieve a structural transition. Furthermore, the experiment suggests an apparent saturation at only 1/3 which could be a result of lead precipitation in the interior lowering the effective c o with annealing time, since the solubility limit of Pb in Ag is below 0.5 at% at 533 K [17]. Consequently, nucleation of the second structure may be generally suppressed at lower temperatures. Moreover, the plateau lengths are rather sensitive to changes in A E and A E., respectively. This is demonstrated in fig. 3, where the kinetics at 564 K is shown for three slightly altered values of A E: Since ('crit . i n c r e ~ w i t h A P a n d qcrit tends closer to co a significantly enlarged delay in formation of the new phase results, although the variation of AE is only in the range of 0.01 eV < k T. Therefore, a respective transition width is expected leading to a smooth increase of further enrichment as observed in AES measurements in contrast to the evaluations with fixed AE. Consequently, the second enrichment rate cannot be used in order to estimate diffusion coefficient and bulk concentration, respectively, as widely done for an undisturbed segregation, e.g. the first enrichment period in the given case [7].
M. MiMzer, J. Wieting / Segregation kSnetics and surface phase transitions
349
3.2. Tin segregation in iron alloys
Similar results as observed for Pb in Ag are measured in Fe-l.9atToSn [8,10]. A n order-disorder transition of segregated tin atoms was found at the (100) surface. A calibration yields surface coverages of 6.1 x 1014 atoms per cm 2 for the ordered c(2 x 2) structure as well as 1.4 x 10 is atoms per cm 2 for final saturation. Therefore, x 0 = 0.45 is evaluated. The. experimental result at T = 858 K indicates an enrichment time of the first structure comparable to the period of the plateau at the apparent saturation level. Fig. 4 shows the result of the given approach with E °> = 1.0 eV, AE = 0.16 eV, A E , = 0.40 eV, where the value of E tl~ is chosen according to suggestions from experiment [18]. The model calculation with these parameters is in agreement with the experimental observation. For a more quantitative comparison we refer to the tin segregation in Fe-6.2at%Si-0.03at%Sn, wbSch is a widely used soft-magnetic material. Zhou et al. [13] investigated the segregation competition of Sn and Si on the (100) surface at 873 K by AES and LEED. Because of Cs°i>> cOn at first silicon segregates and reaches saturation corresponding to the equilibrium in a binary F e - S i alloy at half a monolayer, where a c(2 × 2) structure is formed. Later on tin displaces silicon from the surface, since the interracial activity of tin is considerably higher. The segregation free energy of silicon only amounts to 0.64 eV at the given temperature [2]. The final saturation of tin segregation corresponds to a p(1 × 1) structure being equivalent to a tin surface coverage of one monolayer. Because of different saturation values the competition between both segregants cannot be described in a simple regular solution
1.o CSn
0.8 0.6.
n/.
5'oo
1o'oo
15oo
2ooo os, t 6~
Fig. 4. Calculated surface segregation kinetics of Sn in Fe including a structural transition (x o = 0.45, E t~ =1.0 eV, AE = 0.16 eV, AE. = 0.40 eV, c° = 0.019, T= 858 K).
M. Militzer, J. Wieting / Segregation kinetics and surface phase transitions
350
model. Indeed the latter has to be connected with a structural transition in the surface layer as already proposed. For convenience furthar approximations are possible. Because of its high mobility silicon is in a current equilibrium (qsi = Cs°i) during the replacement. Then tin diffusion determines the competition kinetics. In order to estimate proper local equilibrium conditions for tin segregation a strong repulsive interaction ,,o) ~ ' S i S n between segregated Si and Sn has to be taken into account in the silicon-rich structure 1, whereas self-interactions a~f ) are neglected. The silicon-tin interaction in iron, =
- (
-- , o
-- ,o
os. ) ,
(11)
is expected to be strongly negative from phase diagrams of the respective binary systems. The regular miscibility parameters are given by
toij = Z [ ~.ij - I ( ~.ii -~ ¢£jj)] ,
(12)
where Z denotes the coordination number and % < 0 the pair interactions. Because of the lack of any miscibility between silicon and tin t%iSn > 0 holds, whereas the formation of intermetallic compounds yields evesi ~<0 as well as ~0veS. < 0. Thus a strong repulsive interaction, e.g. ,~(~) "~SiSn --" - 0 . 5 eV, is assumed leading to an extended miscibility gap in the first surface structure at the given temperature. Demixing already occurs for Cs, < 0.01 provided, Csi is sufficiently high. Therefore, the local equilibrium condition of Sn is given up to the structural transition by qSn(t)
=
exp
kT
~'Sn
=q0,
(13)
as shown in ref. [12]. A sufficient tin enrichment enables the transition c(2 × 2) --* p(1 x 1). For structure 2, p(1 x 1), an effective ideal solution model is used, where the strong Si-Sn repulsion is taken into account by an extremely small segregation energy of silicon Es(2) << E Si(1)' whereas ~-t2)> r.tl) "Sn J-'Sn is expected. Thus tin segregation is undisturbed by silicon in the second phase nucleating at t, and qs,(t) is then given by qsn(t) = q ( t ) =
Cs"(t)
1 - Y'.cj(t)
e x p ( E(s2'/kT)
(14)
J
with Csi (t) = [ 1 - Cs. (t)] c°i exp( E(s2'/k T) 1 + c°i exp( E(s2'/k T ) '
(I 5)
characterizing the current equilibrium of silicon in the new structure. Because of a high enrichment ratio of tin in this second phase q ( t ) << cO,
M. Militzer, J. Wieting / Segregation kinetics and surface phase transitions
351
X h.-~',,,,~',='
Fe - 6.2%Si - 0,03%Sn 873K
c,° 1
0#.
~ Si
i,
I ~t/
Sn~~x
zx z~
o.~-
~
x
1o
~ - r -Io~ " t /rnin
Fig. 5. C o m p a r i s o n of the calculated segregation kinetics to the (100) surface in an F e - 6 . 2 a t % S i - 0 . 0 3 a t % S n single crystal at 873 K with A E S measurements (Si: (zx), Sn: ( x ) ) of Z hou et al. [13]; ~t~°),.si= 0.64 eV, E 12)si= 0.1 eV, ---s.~'°)= 1.07 eV, E t2}sn= 1.5 eV, D~.8 -2. = 66).
holds up to final saturation. Therefore, segregation equation (1) is approximated by 2(C~n--q°)(Dsnt//~
Csn(t ) =
2
~r)
!/2
,
t
2(Dsn//82~r),/2(cOnt,/2_ q°t~/2),
(16)
t > t,,
where according to the experiment Csn(0) = 0 is introduced. A fit with the experimental result of Zhou et al. yields Dsn/82= 66 which agrees well with expectations from other experimental results on tin diffusion in ferromagnetic a-Fe [8,19]. It was already mentioned that the segregation free energy of tin in pure iron amounts to about 1.0 eV, in iron-silicon alloys it is expected to be even higher. Using ~'t~) 1.07 eV, L'Sn ~,t2) 1.5 eV as well as Es~i2~= 0.1 eV good agreement with the AES measurement by Zhou et ai. is achieved as shown in fig. 5. For comparison the AES results given in relative peak height ratios (PHR~) with respect to the 703 eV iron signal are converted into coverages by the linear relation ~Sn
"-"
- - "
c, = h i PHIL,
(17)
where the calibration constants h i are deduced from LEED measurements of the corresponding structures
PHRsi.max= A
PHRsn.max
A
¢Si,max "- 0 . 5 , m
= ¢Sn,max --
1
.
For silicon a background signal according to c°i is taken into account.
M. Mifitzer, J. Wieting / Segregation kinetics and surf,zce phase transitions
352
Both experimental results and model calculations show a retarded enrichment rate for tin at the beginning of the segregation process due to the strong repulsion with silicor., which increases significantly during the formation of the second structure. It should be pointed out that the replacement of Si by Sn only occurs, if the tin bulk concentration exceeds q0. Consequently, a proper knowledge of this value is useful in a detailed consideration of the corresponding segregation competition. Moreover, it is worth noting th~,t tin segregation structures in iron alloys completely change with varying bulk composition. Whereas in F e - S n a c(2 x 2) structure and finally a disordered one with a tin coverage of more than one monolayer is found, in Fe-Si tin segregation forms a p(1 x 1) structure at (100) planes.
4. Conduding remarks An improved phenomenological approach including 2D structural transitions explains successfully segregation kinetics in binary as well as ternary alloys. Especially in the latter a more quantitative description of competition effects may be achieved allowing a detailed discussion of conditions being necessary for segregation dominance of distinct species. Moreover, it is possible to deal not only with segregation in metals but also in ceramics as well as semiconductors within the presented theory. Frequently, there is a lack in the knowledge of exact absolute values of segregation parameters (E~tx), aij(x) ). Applying the given approach to respective experimental results it enables a determination of relations between these parameters, e.g. in binary systems the energy differences AE of different structures as well as in ternary systems the relation E~/X)= E~ ~) - E ) x) for competitive segregants. Thus starting from one known value, several segregation parameters may be deduced by this method.
References [11 S. Hofmann, in: Scanning Electron Microscopy/1985, Ed. O. Johari (AMF O'Hare, Chicago, 1985) p. 1071. f'~ll tr..l
l,J,
'm~,,.Jlt~ul~.l.,,
~,JL~.,t,,,I
l%~,,.0..Jr
~.LJt..P,J~
•
l~v.
[31 M.P. Seah, in: Practical Surface Analysis by Auger and X-ray Photoelectron Spectroscopy, Eds. D. Briggs and M.P. Seah (Wiley, New York, 1983) p. 247. 14l C.3. MeMahon, Jr. and L. Marchut, J. Vacuum Sci. Technol. 15 (1978) 450. [51 M.P. Seah and C. Lea, Phil. Mag. 35 (1977) 213. 161 G. Luckman, L.R. Adler and W.R. Graham, Surface Sci. 121 (1982) 61. [71 A. Rolland and J. Bernardini, Scripta Met. 19 (1985) 839. [81 K. Hennesen, H. Keller and H. Viefhaus, Scripta Met. 18 (1984) 1319. [91 A. Rolland, J. Bernardini and M.G. Barthes-Labrousse, Surface Sci. 143 (1984) 579. [101 H. Viefhaus and M. Riisenberg, Surface Sci. 159 (1985) 1.
M. Mifitzer, J. Wieting / Segregation kinetics and surface phase transitions [11] [12] [13] [14] [15] [16] [17]
353
F. Bezuidenhout, J. Du Plessis and P.E. Viljoen, Surface Sci. 171 (1986) 392. M. Militzer and J. Wieting, Acta Met. 35 (1987) 2765. Y.X. Zhou, C.J. McMahon, Jr. and E.W. Plummer, J. Vacuum Sci. Technol. A2 (10~4) 1118. M. Militzer and J. Wieting, Acta Met. 34 (1986) 1229. M. Guttmann, Surface Sci. 53 (1975) 213. P. Haasen, Physikalische Metallkunde (Akademie-Vedag, Berlin, 1985). M. Hansen and K. Anderko, Constitution of BinaD" Alloys (McGraw-Hill, New York, 1958) p. 40. [18] M.P. Seah and C. Lea, Phil. Mag. 31 (1975) 627. [19] W. Arabczyk, M. Militzer, H.J. Miissig and J. Wieting, Scripta Met. 20 (1986) 1549.